Eternal Inflation
The mechanism through which primordial inhomogeneities are produced in
inflation is truly remarkable. Simple
models of inflation involve a scalar field φ
rolling down the hill provided by its assumed potential energy function V(φ). As long as V(φ) is positive, it acts like a
positive cosmological constant which causes exponential expansion of the
Universe. However, the field φ, like every other field, is subject
to quantum mechanical fluctuations. As it rolls down the hill, in some regions φ fluctuates downwards and in others
it fluctuates upwards. The former
regions reach the bottom of the hill, and convert the energy stored in φ into hot radiation sooner: the
latter regions do so later. Since the
density of the radiation rapidly decreases as the universe expands, the regions
undergoing heating later are left with a higher energy density than those
undergoing heating sooner. This is a very beautiful idea, because it links
gravity to quantum mechanics and in fact provides one of our few observational
probes of quantum gravity. All the more
remarkable is that it appears to be consistent with the latest observational
data. If
this is the correct explanation of the origin of structure in the Universe,
then quantum processes were crucial to the formation of the Universe we now see.
The naive approach to inflation begs the question of why the scalar
field started out up the hill. Of course that was necessary in order to get inflation
going, but assuming that is really just assuming the desired answer. The
mechanism of quantum fluctuations offers an intriguing way out of this impasse
which has been seized upon by Guth and others.
The idea is that those regions in which the field jumps uphill will not
only undergo heating later, but they will actually undergo inflation for
longer. Since inflation is
exponentially powerful, regions undergoing inflation for longer expand to
vastly greater size. The thought is therefore that in large parts of the
Universe they will have quantum jumped uphill, and these parts will multiply. Within
these exponentially growing regions, small ‘island Universes’ will appear in
which the field does roll down the hill and releases its energy into hot radiation. The picture is then something like a steady
state universe ‘in the large’, where uphill quantum fluctuations continously
replenish the inflating regions, so that the production of ‘island Universes’
continues indefinitely.
Figure
1: De Sitter spacetime is a four dimensional
hyperboloid embedded in five
dimensional Minkowski spacetime. A two dimensional version is illustrated,
embedded in threedimensional Minkowski spacetime. It may be timesliced on closed, flat or open slices as shown. In
fourdimensional de Sitter spacetime the closed slices are three spheres and
the open slices are three dimensional hyperbolic spaces, or infinite open
universes. The heavy vertical line (dotted
on the rear side) represents the future light cone of a point on the equator
Σ, and is the spacetime locus of a bubble wall nucleated there. Inside the
bubble there is an infinite inflating open universe.
Figure
2: Causal diagram showing the formation of
‘bubble universes’ within de Sitter space. de Sitter spacetime is represented
as the surface of a two dimensional cylinder (here seen from the side) where
the circular cross sections are actually three spheres. The special three sphere Σ is the equator
or initial condition surface for de Sitter space. The point O represents the location of an observer
inside one of the bubble Universes. The past light cone does not intersect any
of the other bubble Universes.
We do not have the techniques to fully verify this behavior not least
because there is no consistent theory of quantum gravity we can use to perform
the calculation. But in spite of this there are various approximate
calculations we can perform and there is a fair amount of evidence that the basic
physical picture is correct. Jumps of φ
uphill lead to exponential expansion because they increase the potential energy
V(φ) increasing its
repulsive effect.
Nevertheless I believe there is a serious flaw in the reasoning, which
renders the eternal inflation effect irrelevant. The problem is that the argument neglects the constraint of
causality.
We are all used to the fact that an event can only be said to be
‘caused’ by something which precedes it. In relativity theory, causality is a
stronger constraint because nothing travels faster than the speed of light.
Thus something can only be ‘caused’ by phenomena within a certain distance from
it, roughly the speed of light times the time before the event.
This region of spacetime is called the past light cone of the
event. All the laws of physics we know
(general relativity and quantum field theory) are consistent with causality in
this form. Causality implies that all
measurable quantities (more precisely, all correlators) at a set of spacetime
points {x} are fully determined by the complete set of measureable quantities
(equivalently, correlators) evaluated on a spacelike surface which fills the
past light cones of all of the points {x} in question.
Figures (1) and (2) show how causality works in an inflating spacetime.
The solution of the field equations in general relativity with a positive
cosmological constant is called de Sitter spacetime. It has the geometry of a four dimensional hyperboloid embedded in
five dimensional Minkowski spacetime Figure (1). For example, if we start the
Universe on a threesphere which is static (i.e. not expanding) then it will
expand due to the repulsive effect of the cosmological constant. The slice
Σ of the spacetime (which is a threesphere) grows exponentially, as we go
forward or backwards in time due to this effect. The diagram also shows that de Sitter spacetime can be
timesliced as a closed Universe, a flat Universe or an open Universe. This
illustrates the point that ‘threevolume’, whilst an intuitive concept, is
actually hard to define. If we speak of our initial inflating region expanding
its volume by some amount, we need to specify which spatial slicing we use. By
changing the slicing, we can completely change the inferred volume, by an
infinite amount.
The second diagram, Figure (2), shows the causal properties of de
Sitter spacetime. In this diagram, physical length and time scales have been
shrunk so that the whole of the spacetime is now a finite region. The infinite
hyperboloid shown in Figure (1) is shrunk to a finite cylinder S^{3} times an
interval. In fact only the region to
the future of the initial surface Σ is shown, for reasons which will be
explained below. The important point however is that in the causal diagram
Figure (2), light rays move at +/ 45^{o} to the vertical.
Now let us consider the production of island hot big bang Universes
within the inflating spacetime. This
description is relatively well understood in the case where the scalar field is
trapped in a metastable minimum of its potential and out of which it quantum
tunnels. The tunneling has the effect of causing bubbles to nucleate inside of
which the field rolls down the potential to the true minimum (assumed to be at
V = 0). Outside the bubbles the field
is nearly constant and the spacetime is near perfect de Sitter spacetime.
The interior of the bubble is the region inside the future lightcone of
the nucleation point. For example, a bubble nucleating on the surface labeled
Σ is bounded by a spherical bubble wall which grows out along the light
cone indicated by the solid vertical line on the right of the diagram. It is a remarkable fact that the surfaces of
constant field Φ, which are also the surfaces of
constant energy density turn out to be the open
slices of de Sitter spacetime. These
slices turn into the natural ‘constant time’ slices in the ensuing hot big
bang. Remarkably therefore, an infinite
open universe is contained in the spacetime future of a perfectly finite
bubble. Thus we see the first emergence of an infinity  the spatial regions undergoing heating following
inflation are in fact infinite open universes. As the lower diagram Figure (2) indicates, a
number of bubbles will nucleate in de Sitter spacetime, in fact an infinite
number since the spacetime volume to the future of Σ is infinite. If the
bubble nucleation rate is small, very few bubbles ever collide. So we have a
picture of an external eternal inflating spacetime which contains an infinite
number of infinite inflating ‘island Universes’, which will heat up and produce
independent hot big bangs.
The existence of these infinities of infinite Universes presents a real
problem, which simply put is the question ‘Where are we?’. It is not clear what
the relative probability is to be in one part of an island Universe as opposed
to another, or in one bubble or another.
This lack of predictivity leads to what Vilenkin has termed the
‘predictability crisis’ in inflation.
Let me expand briefly on what this crisis is. We are only expecting to be able to calculate relative
probabilities for different types of Universes, since that is all quantum
theory allows us to consider. Relative
probabilities are always well defined when the number of distinct possibilities
is finite. For example if one takes a
card from the top of a wellshuffled pack, one expects to get a heart or a club
with equal probabilities. Likewise we
could take many packs and keep more clubs than hearts to bias the probabilities,
but as long as there were a finite total number of cards there would always be
a well defined probability, given a random choice of card.
The situation becomes more difficult when there are an infinite number
of possibilities. We cannot enumerate all the possible orderings and it is hard
to say which are more probable than others. For example, take all the integers
from one to infinity. If we put all the
odd numbers first, it might appear that a number chosen at random would have to
be odd. With the even even numbers put first we would reach the opposite
conclusion (this example has been used by Guth). We need to define the relative probabilities more carefully, and
in such a way that the infinities are taken care of.
It seems to me that the original question of what an eternally inflating
spacetime looks like is actually the wrong question. We should not ask ‘where
are we?’ in the infinite, quantum fluctuaing spacetime. Rather we should note
that causality implies the only parts of the
spacetime that ever influence what we see are the parts inside our past light
cone. An example is shown in Figure (2). An observer at O, inside a bubble, is causally disconnected from
events in bubbles which never collide with that bubble. For an observer living inside a bubble,
everything observable now or in the future should be fully determined by the
physics inside her/his past light cone. It should be completely unnecessary to
discuss the other bubbles. The
constraint of causality is very important because in a stroke it removes the twin problems
of an infinite number of infinite open universes, leaving us with a much better
defined problem.
The question of what an inflating spacetime looks like should therefore
be rephrased. We should ask ‘Given that we have undergone a hot big bang, what
is the most probable past within our past light cone?’
Contributed by: Dr. Neil Turok
